Articles of abstract algebra

For a field $K$, is there a way to prove that $K$ is a PID without mentioning Euclidean domain?

I know that if $K$ is a field then $K[x]$ is a Euclidean domain and every Euclidean domain is a PID. In this way I can prove that $K[x]$ is a PID. But is there a method to show $K[x]$ is a PID directly from the definition? I mean a usual procedure is to design […]

Does Fermat's Little Theorem work on polynomials?

Let $p$ be a prime number. Then if $ f(x) = (1+x)^p$ and $g(x) = (1+x)$, then is $f \equiv g \mod p$? I’m trying to prove that for integers $a > b > 0$ and a prime integer $p$, ${pa\choose b} \equiv {a \choose b}.$ To do this I use FLT to show that […]

Infinite group must have infinite subgroups.

Prove that an Infinite group must have subgroup with infinite elements. I know that if group was cyclic order of the generator is infinite and there are infinite number of divisors.

Are polynomials over $\mathbb{R}$ solvable by radicals?

I know that if $f$ is a polynomial over a subfield $F$ of $\mathbb{C}$ and $f$ is solvable by radicals, then the Galois group of $f$ over $F$ is solvable. I’ve also seen many applications of this fact in demonstrating that certain quintic polynomials over $\mathbb{Q}$ are not solvable by radicals. My question is this: […]

An example about finitely cogenerated modules

We know the fact that: If a module $M$ is finitely cogenerated, then every module that cogenerates $M$ finitely cogenerates $M$. Conversely, it is not true. I find an example in the book “Rings and Categories of modules” written by Frank W. Anderson and Kent R. Fuller. Example The abelian group $\bigoplus_{\mathbb{P}}{\mathbb{Z}_p}$, ($p$ is a […]

Trigonometric solution to solvable equations

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $2, 3, 4 $ are sufficient to express the general solution, for arbitrary degree equations, the general solution can be expressed only in terms of much […]

True/False about ring and integral domain

I have some true or false questions and would like to have your help to check on it. A). in a ring R, if $x^2=x$, $\forall x\in R$, then R is commutative For (A), when looking at $(x+y)^2$, it has $x+y=(x+y)^2=x^2+xy+yx+y^2$ and then yx+xy=0, and from 2x=4x, therefore 2x=0. how this play a role here? […]

Determining the presentation matrix for a module

I am trying to study some module theory using the book “Algebra” by Michael Artin (2nd Edition, to be precise), and I can’t really fathom what is written in Section 14.5. Left multiplication by an $m \times n$ matrix defines a homomorphism of $R$-modules $A: R^n \rightarrow R^m$. Its image consists of all linear combinations […]

Decomposition Theorem for Posets

There is in module theory the following (krull-shmidt) theorem: If $(M_i)_{i\in I}$ and $(N_j)_{j\in J}$ are two families of simple module such that $\bigoplus\limits_{i\in I} M_i \simeq \bigoplus\limits_{j\in J} M_j$, then there exists a bijection $\sigma:I\rightarrow J$ such that $M_i \simeq N_{\sigma(i)}$. Is there a similar kind of theorem for partially ordered sets? More precisely, […]

Proof that $\mathbb{R}$ is not a finite dimensional vector space

How can we prove that the vector space of polynomials in one variable, $\mathbb{R}[x]$ is not finite dimensional?